If you've ever played the classic Windows game, Minesweeper, perhaps you've wondered how geometry and other branches of math can describe the system. If you haven't, here's the Wikipedia entry: Minesweeper. If you have Windows, the game is available on the Mac App Store.
To begin to describe the game, there is a grid of discrete points on various intervals depending on difficulty. There is a randomized function on the grid, which takes each square (or point) of the grid either to an integer between 0 and 8, or a mine. The number is determined by the number of neighboring mines (or, to be mathematical, mines within a neighborhood with radius sqrt(2)). So here's my first difficulty, because it seems here that we're now dealing with two distinct types of coordinates, which I have never seen in math. Furthermore, one type of coordinate depends on proximity to the other. Considering this, it seems tempting to give up altogether the hunt for a mathematically precise description of the game, but let's see what we can come up with.
To begin to describe the game, there is a grid of discrete points on various intervals depending on difficulty. There is a randomized function on the grid, which takes each square (or point) of the grid either to an integer between 0 and 8, or a mine. The number is determined by the number of neighboring mines (or, to be mathematical, mines within a neighborhood with radius sqrt(2)). So here's my first difficulty, because it seems here that we're now dealing with two distinct types of coordinates, which I have never seen in math. Furthermore, one type of coordinate depends on proximity to the other. Considering this, it seems tempting to give up altogether the hunt for a mathematically precise description of the game, but let's see what we can come up with.